Journal of Computer and System Sciences 81 (2015) 661–670 Contents lists available at ScienceDirect Journal of Computer and System Sciences www.elsevier.com/locate/jcss Control complexity in Bucklin and fallback voting: ✩ An experimental analysis ∗ Gábor Erdélyi a, Michael R. Fellows b, Jörg Rothe c, , Lena Schend c a School of Economic Disciplines, University of Siegen, 57076 Siegen, Germany b Parameterized Complexity Research Unit, Charles Darwin University, Darwin, NT 0909, Australia c Institut für Informatik, Heinrich-Heine-Universität Düsseldorf, 40225 Düsseldorf, Germany a r t i c l e i n f o a b s t r a c t Article history: Control in elections models situations in which an external actor tries to change the Received 14 December 2012 outcome of an election by restructuring the election itself. The corresponding decision Received in revised form 30 April 2014 problems have been shown NP-hard for a variety of voting systems. In particular, in our Accepted 5 September 2014 companion paper [16], we have shown that fallback and Bucklin voting are resistant (in Available online 18 November 2014 terms of NP-hardness) to almost all of the common types of control. While NP-hardness Keywords: results for manipulation (another way of tampering with the outcomes of elections) have Computational social choice been challenged experimentally (see, e.g., the work of Walsh [38,37]), such an experimental Control complexity approach is sorely missing for control. We for the first time tackle NP-hard control Bucklin voting problems in an experimental setting. Our experiments allow a more fine-grained analysis Fallback voting and comparison—across various control scenarios, vote distribution models, and voting Experimental analysis systems—than merely stating NP-hardness for all these control problems. Election systems © 2014 Elsevier Inc. All rights reserved. 1. Introduction The notion of control in elections was introduced by Bartholdi et al. [4] to model situations where an external actor, “the chair,” tries to change the outcome of an election by restructuring the election itself. In the constructive control scenario, the chair aims at making a designated candidate the winner of the changed election, while in the destructive control scenario, introduced by Hemaspaandra et al. [23], the chair tries to prevent the current winner from winning. To achieve his goal, the chair might add or delete candidates or voters, or partition them in the course of a control action. Another way of influencing the outcome of an election is manipulation, where a coalition of voters (consisting of one or more voters) participating in the election tries to change the outcome by casting insincere votes, see [3,2,10]. Both scenarios have been studied in terms of their worst-case complexity for various voting systems (see, e.g., the surveys of Faliszewski et al. [20,18], Conitzer [9], and Faliszewski and Procaccia [17]). One of the main issues of this line of research is to determine a natural voting system with a deterministic polynomial-time winner determination procedure that is resistant to as many types of control as possible, where resistance is defined as NP-hardness of the corresponding control ✩ Preliminary versions of parts of this paper appear in the proceedings of the 11th International Symposium on Experimental Algorithms [31] and the COMSOC special session at the 12th International Symposium on Artificial Intelligence and Mathematics [32], and were presented at the Tenth International Meeting of the Society for Social Choice and Welfare and at the Dagstuhl Seminar “Computation and Incentives in Social Choice” [34]. * Corresponding author. E-mail address: [email protected] (J. Rothe). http://dx.doi.org/10.1016/j.jcss.2014.11.003 0022-0000/© 2014 Elsevier Inc. All rights reserved. 662 G. Erdélyi et al. / Journal of Computer and System Sciences 81 (2015) 661–670 problem. As shown in our companion paper [16], fallback voting (a hybrid voting system introduced by Brams and Sanver [6] that combines Bucklin and approval voting) has the broadest resistance to electoral control currently known to hold. To complement these theoretical results, we conducted an experimental analysis of the control complexity in fallback and Bucklin elections, following the approach proposed by Walsh [37,38] that he, Davies et al. [12], and Narodytska et al. [28] (see also [8]) applied to manipulation problems for voting systems such as single transferable vote (STV), veto, Borda’s, Nanson’s, and Baldwin’s rules. They showed that these voting systems can often be manipulated effectively, even though their manipulation problems are NP-hard. Such an experimental approach has been sorely missing for NP-hard control problems; this paper makes the first such attempt. Since both our classical and our parameterized complexity results on control in Bucklin and fallback voting [16] refer to a worst-case measure of complexity, they leave open the possibility that many elections can still be controlled in a reasonable amount of time. While the above-mentioned papers [38,37,12,28] focus on constructive manipulation problems only, we study both constructive and destructive control problems experimentally. When generating random elections in our experiments, we consider two probability distributions: the Impartial Culture model where votes are distributed uniformly and are drawn independently, and the Two Mainstreams model, introduced here to model two mainstreams in society by adapting the Pólya–Eggenberger urn model [5]. In general, our findings indicate that some of the investigated NP-hard control problems can often be solved effectively in practice, whereas for other types of control our experimental results suggest that their problems may indeed be hard to solve even on random instances. Our experiments also allow a more fine-grained analysis than merely stating NP-hardness for all the corresponding control problems. Specifically, we can quantitatively compare constructive with destructive control, control across various voting systems in various control scenarios, and our two particular models of vote distribution. Related work Many of the recent results on the hardness of manipulation problems are concerned with either typical-case analyses and frequency of manipulability1 or quantitative versions of the Gibbard–Satterthwaite theorem; see, in particular, the papers by Mossel et al. [27], Procaccia and Rosenschein [30], Dobzinski and Procaccia [13], Friedgut et al. [21], Zuckerman et al. [42], Xia and Conitzer [39,40], Xia et al. [41], Isaksson et al. [25], Peleg [29], Baharad and Neeman [1], and Slinko [35, 36]. These papers provide theoretical insights into why NP-hard manipulation problems can still be easy to solve in practice. They are complemented by the previously mentioned empirical and experimental studies regarding manipulation problems due to Walsh [38,37], Davies et al. [12,11], and Narodytska et al. [28]. Some of these theoretical and experimental results have been surveyed by Rothe and Schend [32]. To the best of our knowledge, the experimental investigation of NP-hard control problems is new to this paper. Organization We start by giving a brief summary of the theoretical results of control in fallback and Bucklin elections in Section 2, which have been shown in the companion paper [16]. In Section 3 we describe the setup of our experiments including the sampling of randomly generated elections and we describe the implemented algorithms. Section 4 provides a summary of our findings and a discussion of the results before we conclude in Section 5 with open questions and suggestions for future work. 2. Notions and theoretical results from the companion paper [16] Table 1 gives an overview of the theoretical results for control complexity in fallback and Bucklin elections that were shown in the companion paper [16]; we refer to that paper also for the formal definitions of notions mentioned in the table, such as the studied control problems (and the tie-handling rules TE and TP), the notions of immunity, susceptibility, (parameterized) resistance, and vulnerability. Note that, based on the work of Bartholdi et al. [4], Hemaspaandra et al. [23], and Faliszewski et al. [19], 22 types of electoral control have been defined originally and we conducted our experiments for all those 18 types of electoral control Bucklin and fallback voting are not vulnerable to, i.e., the corresponding decision problems are not known to be solvable in deterministic polynomial time. Recently, Hemaspaandra et al. [24] discovered that, depending on the winner model, some of the destructive cases of control by partition of candidates collapse, so there are in fact fewer than 22 distinct types of electoral control. Table 1 indicates these collapses (for the co-winner model) by having only one R entry for two (in fact, identical) control types. 3. Experimental setup In this section we describe the framework of our experimental analysis beginning with the general setup followed by an introduction of the distribution models used to sample the randomly generated elections. We will conclude this section by giving a high-level description of the algorithms. As stated in [16, Section 2] the instances of the problems modeling control by adding or by deleting either candidates or voters contain a parameter k bounding the number of candidates or voters that can be added or deleted, which is crucial 1 Erdélyi et al. [14,15] noted that such approaches—which indeed are very valuable to pursue—are not to be mistaken for work in average-case complexity theory in the sense of Levin [26]. G. Erdélyi et al. / Journal of Computer and System Sciences 81 (2015) 661–670 663 Table 1 Overview of classical and parameterized complexity results for control in Bucklin and fallback voting shown in [16]. All results hold in both the co-winner ∗ and the unique-winner model. Key: I = immune, S = susceptible, R = resistant, R = parameterizedly resistant, V = vulnerable, TE = ties eliminate, and TP = ties promote.